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In physical and organic chemistry, the dispersity is a measure of the heterogeneity of sizes of molecules or particles in a mixture. A collection of objects is called uniform if the objects have the same size, shape, or mass. A sample of objects that have an inconsistent size, shape and mass distribution is called non-uniform. The objects can be in any form of chemical dispersion, such as particles in a colloid, droplets in a cloud,^[1] crystals in a rock,^[2] or polymer molecules in a solvent.^[3] Polymers can possess a distribution of molecular mass; particles often possess a wide distribution of size, surface area and mass; and thin films can possess a varied distribution of film thickness.^{[citation needed]}

IUPAC has deprecated the use of the term polydispersity index having replaced it with the term dispersity, represented by the symbol Đ (pronounced D-stroke^[4]) which can refer to either molecular mass or degree of polymerization. It can be calculated using the equation Đ_M = M_w/M_n, where M_w is the weight-average molar mass and M_n is the number-average molar mass. It can also be calculated according to degree of polymerization, where Đ_X = X_w/X_n, where X_w is the weight-average degree of polymerization and X_n is the number-average degree of polymerization. In certain limiting cases where Đ_M = Đ_X, it is simply referred to as Đ. IUPAC has also deprecated the terms monodisperse, which is considered to be self-contradictory, and polydisperse, which is considered redundant, preferring the terms uniform and non-uniform instead.^[4]

Overview

IUPAC definition

Đ_M = M_w/M_n
where M_w is the weight-average molar mass and
M_n is the number-average molar mass.

Pure Appl. Chem., 2009, 81(2), 351-353

A monodisperse, or uniform, polymer is composed of molecules of the same mass.^[5] Natural polymers are typically monodisperse.^[6] Synthetic monodisperse polymer chains can be made by processes such as anionic polymerization, a method using an anionic catalyst to produce chains that are similar in length. This technique is also known as living polymerization. It is used commercially for the production of block copolymers. Monodisperse collections can be easily created through the use of template-based synthesis, a common method of synthesis in nanotechnology.^{[citation needed]}

A polymer material is denoted by the term polydisperse, or non-uniform, if its chain lengths vary over a wide range of molecular masses. This is characteristic of man-made polymers.[1]. Natural organic matter produced by the decomposition of plants and wood debris in soils (humic substances) also has a pronounced polydispersed character. It is the case of humic acids and fulvic acids, natural polyelectrolyte substances having respectively higher and lower molecular weights. Another interpretation of polydispersity index is explained in the article Dynamic light scattering (cumulant method subheading). In this sense, the PDI values are in the range from 0 to 1.

The polydispersity index (PDI) or heterogeneity index, or simply dispersity (Đ), is a measure of the distribution of molecular mass in a given polymer sample. The PDI calculated is the weight average molecular weight ( $M_{w}$ ) divided by the number average molecular weight ( $M_{n}$ ). It indicates the distribution of individual molecular masses in a batch of polymers. The PDI has a value equal to or greater than 1, but as the polymer chains approach uniform chain length, the PDI approaches unity (1).^[7] For some natural polymers PDI is almost taken as unity. The PDI from polymerization is often denoted as:

$\ PDI=M_{w}/M_{n}$ ,

where $M_{w}$ is the weight average molecular weight and $M_{n}$ is the number average molecular weight. $M_{n}$ is more sensitive to molecules of low molecular mass, while $M_{w}$ is more sensitive to molecules of high molecular mass.

Effect of polymerization mechanism

Typical dispersities vary based on the mechanism of polymerization and can be affected by a variety of reaction conditions. In synthetic polymers, it can vary greatly due to reactant ratio, how close the polymerization went to completion, etc. For typical addition polymerization, Đ can range around 10 to 20. For typical step polymerization, most probable values of Đ are around 2 —Carothers' equation limits Đ to values of 2 and below.

Living polymerization, a special case of addition polymerization, leads to values very close to 1. Such is the case also in biological polymers, where the dispersity can be very close or equal to 1, indicating only one length of polymer is present.

Determination methods

Gel permeation chromatography (also known as size exclusion chromatography)
Light scattering measurements such as dynamic light scattering
Direct measurement via mass spectrometry using MALDI or ESI-MS

References

^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1088/1748-9326/4/1/015002, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1088/1748-9326/4/1/015002 instead.
^ Higgins, Michael D. (2000). "Measurement of crystal size distributions" (PDF). American Mineralogist. 85: 1105–1116.
^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1021/j100847a053, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1021/j100847a053 instead.
^ ^a ^b Stepto, R. F. T.; Gilbert, R. G.; Hess, M.; Jenkins, A. D.; Jones, R. G.; Kratochvíl P. (2009). "Dispersity in Polymer Science" Pure Appl. Chem. 81 (2): 351–353. DOI:10.1351/PAC-REC-08-05-02.
^ "monodisperse polymer (See: uniform polymer)". IUPAC Gold Book. International Union of Pure and Applied Chemistry. Retrieved 25 January 2012.
^ Brown, William H.; Foote, Christopher S.; Iverson, Brent L.; Anslyn, Eric V. (2012). Organic chemistry (6 ed.). Cengage Learning. p. 1161. ISBN 978-0-8400-5498-2.
^ Peter Atkins and Julio De Paula, Atkins' Physical Chemistry, 9th edition (Oxford University Press, 2010, ISBN 978-0-19-954337-3)

External links

Polymer structure

[1] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1088/1748-9326/4/1/015002, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1088/1748-9326/4/1/015002 instead.

[2] Higgins, Michael D. (2000). "Measurement of crystal size distributions" (PDF). American Mineralogist. 85: 1105–1116.

[3] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1021/j100847a053, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1021/j100847a053 instead.

[Stepto-4] Stepto, R. F. T.; Gilbert, R. G.; Hess, M.; Jenkins, A. D.; Jones, R. G.; Kratochvíl P. (2009). "Dispersity in Polymer Science" Pure Appl. Chem. 81 (2): 351–353. DOI:10.1351/PAC-REC-08-05-02.

[5] "monodisperse polymer (See: uniform polymer)". IUPAC Gold Book. International Union of Pure and Applied Chemistry. Retrieved 25 January 2012.

[6] Brown, William H.; Foote, Christopher S.; Iverson, Brent L.; Anslyn, Eric V. (2012). Organic chemistry (6 ed.). Cengage Learning. p. 1161. ISBN 978-0-8400-5498-2.

[7] Peter Atkins and Julio De Paula, Atkins' Physical Chemistry, 9th edition (Oxford University Press, 2010, ISBN 978-0-19-954337-3)

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@@ Line 22: / Line 22: @@
  |align = right
 }}
-A uniform [[polymer]] (formerly described as a monodisperse polymer) is composed of molecules of the same mass.<ref>{{cite web
+A monodisperse, or uniform, [[polymer]] is composed of molecules of the same mass.<ref>{{cite web
 |url=http://goldbook.iupac.org/M04012.html
 |title= monodisperse polymer (See: uniform polymer)
 |work= IUPAC Gold Book
 |publisher= International Union of Pure and Applied Chemistry
-|accessdate=25 January 2012}}</ref> Natural polymers are typically uniform.<ref>{{ cite book
+|accessdate=25 January 2012}}</ref> Natural polymers are typically monodisperse.<ref>{{ cite book
 |last1 = Brown
 |first1 = William H.
@@ Line 42: / Line 42: @@
 |year = 2012
 |isbn = 978-0-8400-5498-2
-|page = 1161 }}</ref> Synthetic uniform polymer chains can be made by processes such as [[anionic]] polymerization, a method using an anionic [[catalyst]] to produce chains that are similar in length. This technique is also known as [[living polymerization]]. It is used commercially for the production of [[block copolymer]]s.  Uniform collections can be easily created through the use of template-based synthesis, a common method of synthesis in [[nanotechnology]].{{Citation needed|date=February 2012}}
+|page = 1161 }}</ref> Synthetic monodisperse polymer chains can be made by processes such as [[anionic]] polymerization, a method using an anionic [[catalyst]] to produce chains that are similar in length. This technique is also known as [[living polymerization]]. It is used commercially for the production of [[block copolymer]]s.  Monodisperse collections can be easily created through the use of template-based synthesis, a common method of synthesis in [[nanotechnology]].{{Citation needed|date=February 2012}}
-A polymer material is denoted by the term  non-uniform (formerly 'polydisperse') if its chain lengths vary over a wide range of molecular masses. This is characteristic of man-made polymers.[http://www.chemicool.com/definition/polydisperse.html]. [[Natural organic matter]] produced by the decomposition of plants and wood debris in soils ([[humic substances]]) also has a pronounced non-uniform character. It is the case of [[humic acid]]s and [[fulvic acid]]s, natural [[polyelectrolyte]] substances having respectively higher and lower molecular masses.  A rather different meaning of polydispersity index is explained in the article [[Dynamic light scattering]] (cumulant method subheading). In this sense, the PDI values are in the range from 0 to 1.
+A polymer material is denoted by the term polydisperse, or non-uniform, if its chain lengths vary over a wide range of molecular masses. This is characteristic of man-made polymers.[http://www.chemicool.com/definition/polydisperse.html]. [[Natural organic matter]] produced by the decomposition of plants and wood debris in soils ([[humic substances]]) also has a pronounced polydispersed character. It is the case of [[humic acid]]s and [[fulvic acid]]s, natural [[polyelectrolyte]] substances having respectively higher and lower molecular weights.  Another interpretation of polydispersity index is explained in the article [[Dynamic light scattering]] (cumulant method subheading). In this sense, the PDI values are in the range from 0 to 1.
+The '''polydispersity index''' ('''PDI''') or heterogeneity index, or simply dispersity (Đ), is a measure of the distribution of [[molecular mass]] in a given [[polymer]] sample.  The PDI calculated is the [[weight average molecular weight]] (<math> M_w </math>) divided by the [[number average molecular weight]] (<math> M_n </math>).  It indicates the distribution of individual [[molecular mass]]es in a batch of [[polymer]]s.  The PDI has a value equal to or greater than 1, but as the polymer chains approach uniform chain length, the PDI approaches unity (1).<ref>Peter Atkins and Julio De Paula, ''Atkins' Physical Chemistry'', 9th edition (Oxford University Press, 2010, ISBN 978-0-19-954337-3)</ref> For some natural polymers PDI is almost taken as unity.
+The PDI from polymerization is often denoted as:
+<math>\ PDI = M_w/M_n</math>,
-The dispersity (''Đ'', formerly known as '''polydispersity index''', '''PDI''', or heterogeneity index), is a measure of the distribution of [[molecular mass]] in a given [[polymer]] sample.  The dispersity calculated is the [[weight average molecular weight]] (<math> M_w </math>) divided by the [[number average molecular weight]] (<math> M_n </math>).  It indicates the distribution of individual [[molecular mass]]es in a batch of [[polymer]]s.  The dispersity has a value equal to or greater than 1, but as the polymer chains approach uniform chain length, ''Đ'' approaches unity (1).<ref>Peter Atkins and Julio De Paula, ''Atkins' Physical Chemistry'', 9th edition (Oxford University Press, 2010, ISBN 978-0-19-954337-3)</ref> For some natural polymers, ''Đ'' is almost taken as unity.
-The dispersity from polymerization is often denoted as ''Đ'' <math> = M_w/M_n</math>,
 where <math> M_w </math> is the [[weight average molecular weight]]  and  <math> M_n </math> is the [[number average molecular weight]]. <math> M_n </math> is more sensitive to molecules of low molecular mass, while <math> M_w </math> is more sensitive to molecules of high molecular mass.
 == Effect of polymerization mechanism ==
-Typical dispersities vary based on the mechanism of polymerization and can be affected by a variety of reaction conditions.  In synthetic polymers, it can vary greatly due to [[reactant]] ratio, how close the [[polymerization]] went to completion, etc.  For typical addition [[polymerization]], ''Đ'' can range around 10 to 20.  For typical step polymerization, most probable values of ''Đ'' are around 2 &mdash;[[Carothers' equation]] limits ''Đ'' to values of 2 and below.
+Typical dispersities vary based on the mechanism of polymerization and can be affected by a variety of reaction conditions.  In synthetic polymers, it can vary greatly due to [[reactant]] ratio, how close the [[polymerization]] went to completion, etc.  For typical addition [[polymerization]], Đ can range around 10 to 20.  For typical step polymerization, most probable values of Đ are around 2 &mdash;[[Carothers' equation]] limits Đ to values of 2 and below.
 [[Living polymerization]], a special case of addition polymerization, leads to values very close to 1.  Such is the case also in biological polymers, where the dispersity can be very close or equal to 1, indicating only one length of polymer is present.